Multiply the following complex numbers, marked as blue dots on the graph: $[3(\cos(\frac{1}{6}\pi) + i \sin(\frac{1}{6}\pi))] \cdot [3(\cos(\frac{4}{3}\pi) + i \sin(\frac{4}{3}\pi))]$ (Your current answer will be plotted in orange.)
Solution: Multiplying complex numbers in polar forms can be done by multiplying the lengths and adding the angles. The first number ( $3(\cos(\frac{1}{6}\pi) + i \sin(\frac{1}{6}\pi))$ ) has angle $\frac{1}{6}\pi$ and radius $3$ The second number ( $3(\cos(\frac{4}{3}\pi) + i \sin(\frac{4}{3}\pi))$ ) has angle $\frac{4}{3}\pi$ and radius $3$ The radius of the result will be $3 \cdot 3$ , which is $9$ The angle of the result is $\frac{1}{6}\pi + \frac{4}{3}\pi = \frac{3}{2}\pi$ The radius of the result is $9$ and the angle of the result is $\frac{3}{2}\pi$.